Explicit expression for inverse of an automorphism of a power series ring

Calculating the inverse of an automorphism of a formal power series ring presents a frequent challenge in a myriad of mathematical inquiries, especially in the realm of singularity theory. In instances involving non-linear and multivariable contexts, S. S. Abhyankar pioneered a methodology to tackle this problem. However, calculating the expressions up to a certain order using this method requires calculating higher-order terms and then carry out the selection, which leads to redundant computations in practice. This article introduces two novel approaches for determining the inverse of an automorphism of a formal power series ring over an arbitrary commutative ring with unit, grounded in the newly developed higher order Jacobian matrix theory. These approaches can be conceived as non-linear extensions of the inverse matrix method and the Gaussian elimination method respectively. They avoid redundant computations above. For the two new methods, we also give the application in calculating the explicit expression for the implicit function theorem.